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Several Amazing Discoveries About Compact Metrizable Spaces in ZF

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Several Amazing Discoveries About Compact Metrizable Spaces in ZF Several amazing discoveries about compact metrizable spaces in ZF Kyriakos Keremedis, Eleftherios Tachtsis and Eliza Wajch Department of Mathematics, University of the Aegean Karlovassi, Samos 83200, Greece [email protected] Department of Statistics and Actuarial-Financial Mathematics, University of the Aegean, Karlovassi 83200, Samos, Greece [email protected] Institute of Mathematics Faculty of Exact and Natural Sciences Siedlce University of Natural Sciences and Humanities ul. 3 Maja 54, 08-110 Siedlce, Poland [email protected] August 5, 2020 Abstract arXiv:2008.01233v1 [math.GN] 3 Aug 2020 In the absence of the axiom of choice, the set-theoretic status of many natural statements about metrizable compact spaces is investi- gated. Some of the statements are provable in ZF, some are shown to be independent of ZF. For independence results, distinct models of ZF and permutation models of ZFA with transfer theorems of Pincus are applied. New symmetric models are constructed in each of which the power set of R is well-orderable, the Continuum Hypothesis is sat- isfied but a denumerable family of non-empty finite sets can fail to have a choice function, and a compact metrizable space need not be embeddable into the Tychonoff cube [0, 1]R. 1 Mathematics Subject Classification (2010): 03E25, 03E35, 54A35, 54E35, 54D30 Keywords: Weak forms of the Axiom of Choice, metrizable space, to- tally bounded metric, compact space, permutation model, symmetric model. 1 Preliminaries 1.1 The set-theoretic framework In this paper, the intended context for reasoning and statements of theorems is the Zermelo-Fraenkel set theory ZF without the axiom of choice AC. The system ZF + AC is denoted by ZFC. We recommend [32] and [33] as a good introduction to ZF. To stress the fact that a result is proved in ZF or ZF + A (where A is a statement independent of ZF), we shall write at the beginning of the statements of the theorems and propositions (ZF) or (ZF + A), respectively. Apart from models of ZF, we refer to some models of ZFA (or ZF0 in [15]), that is, we refer also to ZF with an infinite set of atoms (see [20], [21] and [15]). Our theorems proved here in ZF are also provable in ZFA; however, we also mention some theorems of ZF that are not theorems of ZFA. We denote by ω the set of all non-negative integers (i.e., finite ordinal numbers of von Neumann). As usual, if n ∈ ω, then n +1 = n ∪ {n}. Members of the set N = ω \{0} are called natural numbers. The power set of a set X is denoted by P(X). A set X is called countable if X is equipotent to a subset of ω. A set X is called uncountable if X is not countable. A set X is finite if X is equipotent to an element of ω. An infinite set is a set which is not finite. An infinite countable set is called denumerable. A cardinal number of von Neumann is an initial ordinal number of von Neumann. If X is a set and κ is a non-zero cardinal number of von Neumann, then [X]κ is the family of all subsets of X equipotent to κ, [X]≤κ is the collection of all subsets of X equipotent to subsets of κ, and [X]<κ is the family of all subsets of X equipotent to a cardinal number of von Neumann which belongs to κ. For a set X, we denote by |X| the cardinal number of X in the sense of Definition 11.2 of [20]. We recall that, in ZF, for every set X, the cardinal number |X| exists; however, X is equipotent to a cardinal number of von Neumann if and only if X is well-orderable. For sets X and Y , the inequality |X|≤|Y | 2 means that X is equipotent to a subset of Y . The set of all real numbers is denoted by R and, if it is not stated other- wise, R and every subspace of R are considered with the usual topology and with the metric induced by the standard absolute value on R. 1.2 Notation and basic definitions In this subsection, we establish notation and recall several basic definitions. Let X = hX,di be a metric space. The d-ball with centre x ∈ X and radius r ∈ (0, +∞) is the set Bd(x, r)= {y ∈ X : d(x, y) <r}. The collection 1 τ(d)= {V ⊆ X :(∀x ∈ V )(∃n ∈ ω)B (x, ) ⊆ V } d 2n is the topology in X induced by d. For a set A ⊆ X, let δd(A)=0 if A = ∅, and let δd(A) = sup{d(x, y): x, y ∈ A} if A =6 ∅. Then δd(A) is the diameter of A in X. Definition 1.1. Let X = hX,di be a metric space. (i) Given a real number ε > 0, a subset D of X is called ε-dense or an ε-net in X if X = Bd(x, ε). xS∈D (ii) X is called totally bounded if, for every real number ε> 0, there exists a finite ε-net in X. (iii) X is called strongly totally bounded if it admits a sequence (Dn)n∈N N 1 X such that, for every n ∈ , Dn is a finite n -net in . (iv) (Cf. [24].) d is called strongly totally bounded if X is strongly totally bounded. Remark 1.2. Every strongly totally bounded metric space is evidently totally bounded. However, it was shown in [24, Proposition 8] that the sentence “Every totally bounded metric space is strongly totally bounded” is not a theorem of ZF. 3 Definition 1.3. Let X = hX, τi be a topological space and let Y ⊆ X. Suppose that B is a base of X. (i) The closure of Y in X is denoted by clτ (Y ) or clX(Y ). (ii) τ|Y = {U ∩ Y : U ∈ τ}. Y = hY, τ|Y i is the subspace of X with the underlying set Y . (iii) BY = {U ∩ Y : U ∈ B}. Clearly, in Definition 1.3 (iii), BY is a base of Y. In Section 5, it is shown that BY need not be equipotent to a subset of B. In the sequel, boldface letters will denote metric or topological spaces (called spaces in abbreviation) and lightface letters will denote their under- lying sets. Definition 1.4. A collection U of subsets of a space X is called: (i) locally finite if every point of X has a neighbourhood meeting only finitely many members of U; (ii) point-finite if every point of X belongs to at most finitely many mem- bers of U; (iii) σ-locally finite (respectively, σ-point-finite) if U is a countable union of locally finite (respectively, point-finite) subfamilies. Definition 1.5. A space X is called: (i) first-countable if every point of X has a countable base of neighbour- hoods; (ii) second-countable if X has a countable base. Given a collection {Xj : j ∈ J} of sets, for every i ∈ J, we denote by πi the projection πi : Xj → Xi defined by πi(x) = x(i) for each x ∈ Xj. jQ∈J jQ∈J If τj is a topology in Xj, then X = Xj denotes the Tychonoff product of jQ∈J the topological spaces Xj = hXj, τji with j ∈ J. If Xj = X for every j ∈ J, J then X = Xj. As in [8], for an infinite set J and the unit interval [0, 1] jQ∈J of R, the cube [0, 1]J is called the Tychonoff cube. If J is denumerable, then 4 the Tychonoff cube [0, 1]J is called the Hilbert cube. In [12], all Tychonoff cubes are called Hilbert cubes. In [42], Tychonoff cubes are called cubes. We recall that if Xj =6 ∅, then it is said that the family {Xj : j ∈ J} jQ∈J has a choice function, and every element of Xj is called a choice function jQ∈J of the family {Xj : j ∈ J}. A multiple choice function of {Xj : j ∈ J} is every function f ∈ P(Xj) such that, for every j ∈ J, f(j) is a non- jQ∈J empty finite subset of Xj. A set f is called partial (multiple) choice function of {Xj : j ∈ J} if there exists an infinite subset I of J such that f is a (multiple) choice function of {Xj : j ∈ I}. Given a non-indexed family A, we treat A as an indexed family A = {x : x ∈ A} to speak about a (partial) choice function and a (partial) multiple choice function of A. Let {Xj : j ∈ J} be a disjoint family of sets, that is, Xi ∩ Xj = ∅ for each pair i, j of distinct elements of J. If τj is a topology in Xj for every j ∈ J, then Xj denotes the direct sum of the spaces Xj = hXj, τji with j ∈ J. jL∈J Definition 1.6. (Cf. [2], [34] and [26].) (i) A space X is said to be Loeb (respectively, weakly Loeb) if the family of all non-empty closed subsets of X has a choice function (respectively, a multiple choice function). (ii) If X is a (weakly) Loeb space, then every (multiple) choice function of the family of all non-empty closed subsets of X is called a (weak) Loeb function of X. Other topological notions used in this article but not defined here are standard. They can be found, for instance, in [8] and [42]. Definition 1.7. A set X is called: (i) a cuf set if X is expressible as a countable union of finite sets (cf.
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